Keywords

Ordered uniform space, Coincident point, Common fixed point, Compatible pair of mappings, Altering distance function, Generalized weakly C-contraction.

Introduction

The well known Banach fixed point theorem for contraction mapping has been generalized and extended in many directions. Since the uniform spaces form a natural extension of the metric spaces, there exists a considerable literature of fixed point theory dealing with results on fixed or common fixed points in uniform spaces.

A new category of fixed point problems was addressed by Khan et al [12].They introduced the notion of an altering distance function which is a control function that alters distance between two points in a metric space.

Definition 1.1 [12]

The function is called an altering distance function, if the following properties are satisfied:

(i) is continuous and non decreasing,

(ii) if and only if .

Altering distance has been used in metric fixed point theory in recent papers [3,6,11,14]. Choudhury [2] also introduced the following definition.

Definition 1.2 [2]

A mapping , where is a metric space is said to be weakly Ccontractive if for all , the following inequality holds:

Where is a continuous function such that if and only if

In [2] the author proves that if X is complete then every weak C- contraction has a unique fixed Point. Also fixed point theorems in partially ordered spaces and sequentially complete Hausdorff ordered uniform spaces are given in [1,4,5,7,9,13].

In this paper we establish some coincidence and common fixed point results for four self mappings on a Hausdorff sequentially complete ordered uniform spaces satisfying a generalized weak Ccontractive condition which involves altering distance function.

Now, we recall some relevant definitions and properties.

We call a (X, U) pair to be a uniform space which consists of a non empty set X together with a uniformity U. It is well known (see Dugundji [8] and Kelley [10] that any uniform structure U on X is induced by a family D of pseudometrics on X and conversely any family D of pseudometrics on a set X induces on X a structure of uniform space U. In addition, U is Hausdorff if and only if D is separating. A family of pseudometrics on X is said to be separating if for each pair of points there is a such that .

Consider a uniform space (X, U) with a uniformity U induced by a family of pseudometrics on X. A sequence of elements in X is said to be Cauchy if for every and , there is an N with for all and . The sequence is called convergent if there exists an such for every and , there is an N with for all . A uniform space is called sequentially complete if any Cauchy sequence is convergent. A subset of X is said to be sequentially closed if it contains the limit of any convergent sequence of its elements.

Let X be a non-empty set, are given self mappings on X. If for some , then x is called a coincidence point of and s, and w is called a point of coincidence of and s. If , then x is called a common fixed point of and s.

Definition 1.3 [7]

Let be a partially ordered set. Two mappings are said to be weakly increasing if and for all .

Let X be a non-empty set and be a given mapping. For every , we denote by and the subset of X defined by:

And

Definition 1.4

Let are given self mappings on X. The pair is said to be compatible if for each , whenever is a sequence in X such that for some .

MAIN RESULT

Theorem 2.1

Let be a sequentially complete Hausdorff ordered uniform space. Let be given mapping satisfying.

(i) ,

(ii) and are sequentially continuous,

(iii) the pairs and are compatible,

(iv) and are weakly increasing with respect to and .

Suppose that for every and such that and are comparable, we have.

Where for each is an altering distance function and is a continuous function with if and only if .

Then and have a coincidence point , that is, .

Proof

Let be an arbitrary point in . Since there exists such that Since , there exists such that Continuing this process, we can construct sequences and in X defined by.

By construction, we have and then using the fact that f and g are weakly increasing with respect to h and k, we obtain.

Then Or

Since and are comparable for each by inequality (1), we have.

Since is a non decreasing function, we get that.

By triangular inequality, we have.

…………………………..4 Thus

It follows that the sequence is monotonic decreasing. Hence, there exists such that.

By (4) we have

…………………………………….………9

Taking and using (8), we get-

Taking in (3) and using the continuity of and (8), (11) we get that-

Which implies that and hence , so we have.

To prove that is a Cauchy sequence in X, it is sufficient to show that is a Cauchy sequence. Suppose to the contrary, that is not a Cauchy sequence. Then there exists and for which we can find two subsequences and such that is the smallest index for which,

This means that,

Therefore, we use (13), (14) and triangular inequality to get,

Taking in the above inequality and using (12), we find,

On the other hand, we have,

Taking in the above inequality and using (12), (15) we find,

On the other hand, we have,

Taking in the above inequality and using (12), (15) we have,

Also, by triangular inequality, we have,

Taking again in the above inequality and using (12), (15) and (16) we find,

Similarly, we can show that,

From (1), we have,

Since is a non decreasing function, we get that,

Taking again in the above inequality and using (15), (18) we find,

Therefore, from (17) and (20) we have,

Taking in (19) and using (15), (18), (21) and the continuity of we find and , we get that,

Which implies that and hence =0, a contradiction. Thus is a Cauchy sequence and hence is a Cauchy sequence. Since sequentially complete Hausdorff uniform space, there is such that,

Therefore,

From the sequentially continuity of h and k, we get,

Therefore,

The triangular inequality and (2) yields,

From (2) and (22),

The pair is compatible, then,

Using the sequentially continuity of and (25), we have,

Combining (23), (26) together with (27) and taking in (24), we obtain,

Which means that and . So u is a coincidence point of and .

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